300:, whose Sylow subgroups are cyclic, A-groups can be easier to study than general finite groups because of the restrictions on the local structure. For instance, a more precise enumeration of soluble A-groups was found after an enumeration of
485:
336:
186:
The derived length of an A-group can be arbitrarily large, but no larger than the number of distinct prime divisors of the order, stated in (
127:). Modern interest in A-groups was renewed when new enumeration techniques enabled tight asymptotic bounds on the number of distinct
259:
716:
391:
286:
120:
328:
87:
A-groups. Hall's presentation was rather brief without proofs, but his remarks were soon expanded with proofs in (
173:
156:
721:
671:
228:
92:
606:
562:
16:
This article is about a type of mathematical group. For the third millennium BC Nubian culture, see
111:). The focus on soluble A-groups broadened, with the classification of finite simple A-groups in (
60:
641:
Venkataraman, Geetha (1997), "Enumeration of finite soluble groups with abelian Sylow subgroups",
688:
669:
Walter, John H. (1969), "The characterization of finite groups with abelian Sylow 2-subgroups.",
630:
458:
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442:
372:
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332:
680:
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614:
570:
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434:
403:
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626:
590:
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384:
696:
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622:
586:
542:
491:
477:
469:
450:
411:
380:
166:
100:
610:
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255:
232:
224:
202:
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152:
84:
64:
17:
710:
634:
462:
367:
301:
68:
40:
574:
251:
32:
512:
422:
128:
44:
24:
407:
618:
529:
438:
654:
582:
538:
446:
376:
346:
47:, and are still studied today. A great deal is known about their structure.
353:
Broshi, Aviad M. (1971), "Finite groups whose Sylow subgroups are abelian",
180:
119:). Interest in A-groups also broadened due to an important relationship to
107:). The work of Hall, Taunt, and Carter was presented in textbook form in (
503:
148:
553:
Ol'šanskiĭ, A. Ju. (1969), "Varieties of finitely approximable groups",
692:
394:(1962), "Nilpotent self-normalizing subgroups and system normalizers",
297:
323:
Blackburn, Simon R.; Neumann, Peter M.; Venkataraman, Geetha (2007),
684:
212:
A soluble A-group has a unique maximal abelian normal subgroup (
115:) which allowed generalizing Taunt's work to finite groups in (
285:
All the groups in the variety generated by a finite group are
309:
99:). Carter then published an important relationship between
327:, Cambridge Tracts in Mathematics no 173 (1st ed.),
289:
if and only if that group is an A-group, as shown in (
555:
Izvestiya
Akademii Nauk SSSR. Seriya Matematicheskaya
254:
is an A-group if and only if it is isomorphic to the
43:. The groups were first studied in the 1940s by
427:Journal für die reine und angewandte Mathematik
425:(1940), "The construction of soluble groups",
396:Proceedings of the London Mathematical Society
290:
227:A-group is equal to the direct product of the
124:
308:). A more leisurely exposition is given in (
83:, Sec. 9), where attention was restricted to
79:The term A-group was probably first used in (
8:
305:
132:
304:with fixed, but arbitrary Sylow subgroups (
169:is an A-group if and only if it is abelian.
310:Blackburn, Neumann & Venkataraman 2007
143:The following can be said about A-groups:
528:
366:
162:Every finite abelian group is an A-group.
244:
191:
108:
279:
243:), and presented in textbook form in (
190:), and presented in textbook form as (
116:
112:
104:
39:is a type of group that is similar to
240:
88:
7:
643:The Quarterly Journal of Mathematics
597:Taunt, D. R. (1949), "On A-groups",
236:
213:
206:
187:
80:
507:, especially Kap. VI, §14, p751–760
96:
176:is an A-group that is not abelian.
63:with the property that all of its
14:
575:10.1070/IM1969v003n04ABEH000807
476:(in German), Berlin, New York:
174:symmetric group on three points
1:
599:Proc. Cambridge Philos. Soc.
368:10.1016/0021-8693(71)90044-5
325:Enumeration of finite groups
95:of A-groups was studied in (
517:Nagoya Mathematical Journal
738:
329:Cambridge University Press
278:≡ 3,5 mod 8, as shown in (
15:
619:10.1017/S0305004100000414
530:10.1017/S0027763000023023
439:10.1515/crll.1940.182.206
159:of A-groups are A-groups.
408:10.1112/plms/s3-12.1.535
131:classes of A-groups in (
194:, Kap. VI, Satz 14.16).
655:10.1093/qmath/48.1.107
247:, Kap. VI, Satz 14.8).
199:lower nilpotent series
672:Annals of Mathematics
287:finitely approximable
250:A non-abelian finite
93:representation theory
717:Properties of groups
511:Itô, Noboru (1952),
231:of the terms of the
183:order is an A-group.
103:and Hall's work in (
611:1949PCPS...45...24T
567:1969IzMat...3..867O
239:), then proven in (
235:, first stated in (
201:coincides with the
121:varieties of groups
513:"Note on A-groups"
355:Journal of Algebra
270:> 3 and either
675:, Second Series,
645:, Second Series,
487:978-3-540-03825-2
338:978-0-521-88217-0
306:Venkataraman 1997
256:first Janko group
133:Venkataraman 1997
27:, in the area of
729:
703:
665:
649:(189): 107–125,
637:
593:
549:
532:
506:
474:Endliche Gruppen
465:
418:
398:, Third Series,
392:Carter, Roger W.
387:
370:
349:
221:Fitting subgroup
101:Carter subgroups
29:abstract algebra
737:
736:
732:
731:
730:
728:
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707:
706:
685:10.2307/1970648
668:
640:
596:
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510:
488:
478:Springer-Verlag
468:
421:
390:
352:
339:
322:
319:
291:Ol'šanskiĭ 1969
179:Every group of
167:nilpotent group
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125:Ol'šanskiĭ 1969
77:
65:Sylow subgroups
53:
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12:
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5:
735:
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708:
705:
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679:(3): 405–514,
666:
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561:(4): 915–927,
557:(in Russian),
550:
508:
486:
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419:
388:
350:
337:
318:
315:
314:
313:
302:soluble groups
294:
283:
248:
233:derived series
217:
210:
203:derived series
195:
184:
177:
170:
163:
160:
157:direct product
153:quotient group
140:
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123:discussed in (
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41:abelian groups
18:Nubian A-Group
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722:Finite groups
720:
718:
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473:
430:
426:
423:Hall, Philip
399:
395:
358:
354:
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275:
271:
267:
261:
252:simple group
245:Huppert 1967
192:Huppert 1967
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109:Huppert 1967
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59:is a finite
56:
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36:
33:group theory
22:
470:Huppert, B.
433:: 206–214,
402:: 535–563,
280:Walter 1969
129:isomorphism
117:Broshi 1971
113:Walter 1969
105:Carter 1962
45:Philip Hall
25:mathematics
711:Categories
317:References
312:, Ch. 12).
241:Taunt 1949
139:Properties
89:Taunt 1949
51:Definition
635:120131175
583:0373-2436
539:0027-7630
523:: 79–81,
463:118354698
447:0075-4102
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347:154682311
237:Hall 1940
214:Hall 1940
207:Hall 1940
188:Hall 1940
181:cube-free
165:A finite
81:Hall 1940
31:known as
472:(1967),
298:Z-groups
225:solvable
149:subgroup
97:Itô 1952
91:). The
701:0249504
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607:Bibcode
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563:Bibcode
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455:0002877
416:0140570
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229:centers
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75:History
69:abelian
57:A-group
37:A-group
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